Digital filtering involves manipulating one or more digital input signals in order to provide one or more digital output signals. Digital signals are a series of digital values wherein each valve represents the magnitude of a particular signal, such as an electrical signal, at a particular time. In the case of electrical signals, a digital input signal can be produced by sampling the electrical signal periodically and converting each sampled value of the signal to a digital value. Similarly, a digital output signal can be converted to an analog electrical output signal by converting the series of digital values to analog values.
A digital filter can correspond to a known analog filter. For example, it is possible to construct a digital low-pass filter which operates in a manner equivalent to an analog low-pass filter having a resistor and a capacitor. The output signal of such a digital low-pass filter will be the input signal having high frequency components attenuated therefrom. An example of a digital low-pass filter can be represented by the following equation: EQU Y[n]=(Y[n-1]+X[n])/2
Y[n] represents a digital value indicative of the magnitude of the present value of the digital filter output. Y[n-1] represents a digital value indicative of the magnitude of the most recent digital filter output (i.e. the output of the digital filter for the previous iteration). X[n] represents a digital value indicative of the magnitude of the present input to the digital filter. For the low-pass digital filter represented by the above equation, the magnitude of the output equals one half of the sum of the magnitudes of the next most recent output and the input.
The flexibility of digital filters makes it possible in some instances to adjust the parameters of a filter during operation. For example, if a digital filter is implemented using computer software, then the equation that governs operation of the filter can be easily modified or replaced, thus changing characteristics of the filter. A digital filter which can be modified during operation is called an "adaptive digital filter". Adaptive digital filters are very useful in applications where the characteristics of the signal being filtered change over time.
A difficulty with adaptive digital filters is establishing criteria for modifying the filters. It is desirable to modify the filters in response to actual changes in the signal and to not modify the filters in response to spurious noise in the system. The modification mechanism needs to be relatively quick in order to respond adequately to actual signal changes. At the same time, the modification mechanism needs to be relatively slow in order to be resistant to spurious noise.
Techniques for theoretically determining the optimum rate at which an adaptive digital filter should be allowed to change are set forth in W. A. Gardner, "Nonstationary Learning Characteristics of the LMS Algorithm," IEEE Trans, on Circuits and Systems, Vol. CAS-34, No. 10, pp. 1199-1207, October 1987 and in B. Widrow, J. M. McCool, M. G. Larimore, C. R. Johnson, "Stationary and Nonstationary Learning Characteristics of the LMS Adaptive Filter," Proc. IEEE, Vol. 64, No. 8, pp. 1151-1162, August 1976. However, the Gardner and Widrow results are only theoretical because they require values of parameters which cannot be measured in real-world applications of adaptive digital filter.
There are also a variety of ad-hoc techniques for determining the optimum rate of change for an adaptive digital filter, such as that shown in U.S. Pat. No. 4,349,889 to van den Elzen et. al titled "NON-RECURSIVE FILTER HAVING ADJUSTABLE STEP-SIZE FOR EACH ITERATION". The majority of these ad hoc techniques adjust the rate according to a measure of the filter error (i.e. the difference between the actual output and the idealized output of the filter). However, this measurement does not distinguish between spurious noise, which should be ignored and hence should cause the rate of change of the adaptive filter to decrease, and an actual change in the signal being tracked, which should cause the rate of change of the adaptive filter to increase.